Volume 9, Issue 3

The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included.

In this paper, we introduce a method to obtain the dual sequence of a given polynomial set using the lowering operator associated with the involved polynomials. As application, we derive polynomial expansions of analytic functions. The particular case corresponding to Boas–Buck polynomials allows us to unify many polynomial expansions of analytic functions in the literature. This method can be useful in studying many problems arising in the theory of polynomials as the so-called connection and linearization problems.

The Banach contraction principle is used to investigate the existence and uniqueness of solutions for first and second order impulsive semilinear neutral functional differential equations in Banach spaces.

We consider a Cauchy problem for nonlinear parabolic equations with functional dependence. We prove convergence theorems for a general quasilinearization method in two cases: (i) the Hale functional acting only on the unknown function, (ii) including partial derivatives of the unknown function.

Using the theory of spline functions, we investigate the problem of minimization of a generalized Dirichlet integral where Ω is a bounded domain of an 𝑛-dimensional Euclidean space 𝑅 𝑛 , λ ≥ 0 is a fixed number, and 𝑢 𝑥 𝒊 is a generalized according to Sobolev with respect to 𝑥 𝒊 derivative of the function 𝑢 defined on Ω. Minimization is realized with respect to the functions 𝑢 whose boundary values on Γ form a preassigned function, and for them 𝐹 λ (𝑢) is finite.

A theorem concerning some descriptive properties of σ -ideals and generalizing the main result of [Brzuchowski, Cichoń, Grzegorek, and Ryll-Nardzewski, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27: 447–448, 1979] is proved. Various applications of this theorem are also presented.

We employ the decomposition theorem with respect to the cyclic group of order 𝑘 to introduce families of pseudo-Bessel functions and to study their properties. We also discuss in brief their applications in probability theory and in electromagnetism.

Given a complex Banach algebra, we consider the Stiefel bundle relative to the similarity class of a fixed projection. In the holomorphic category the Stiefel bundle is a holomorphic locally trivial principal bundle over a certain Grassmann manifold. Our main application concerns the holomorphic parametrization of framings for projections. In the spatial case this amounts to a holomorphic parametrization of framings for a corresponding complex Banach space.

For a not empty perfect subset of the unit circle C there is a perfect subset of C measure zero which being rotated to every position intersects the first set on a nonempty perfect set. This result may be stated in terms of set of distances between pairs of points from these two sets. A generalization of this result to a product of tori is suggested.

A boundary value problem of linear conjugation is considered for more general curves than those studied previously. A condition on the coefficient is found, under which the classical results are valid for these curves.

Nonimprovable in a certain sense, sufficient conditions for the solvability and unique solvability of the problem 𝑢′ (𝑡) = 𝐹 (𝑢) (𝑡), 𝑢(𝑎) – λ𝑢(𝑏) = ℎ(𝑢) are established, where 𝐹 : 𝐶([𝑎, 𝑏]; 𝑅) → 𝐿([𝑎, 𝑏];𝑅) is a continuous operator satisfying the Carathéodory condition, ℎ : 𝐶([𝑎, 𝑏]; 𝑅) → 𝑅 is a continuous functional, and λ ∈ 𝑅 + .

In the category of differential algebras with strong homotopy there is a Gugenheim's map { ρ 𝑖 } : 𝐴* → 𝐶* from Sullivan's commutative cochain complex to the singular cochain complex of a space, which induces a differential graded coalgebra map of appropriate Bar constructions. Both (𝐵𝐴*, d BA , Δ,) and (𝐵𝐶*, d BC* , Δ,) carry multiplications. We show that the Gugenheim's map 𝐵{ ρ 𝑖 } : (𝐵𝐴*, d BA* , Δ,) → (𝐵𝐶*, d BC* , Δ,) is multiplicative up to homotopy with respect to these structures.

The author first establishes the frame characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. As applications, the author then obtains some estimates of entropy numbers for the compact embeddings between Besov spaces or between Triebel–Lizorkin spaces. Moreover, some real interpolation theorems on these spaces are also established by using these frame characterizations and the abstract interpolation method.

In this paper, most of classical and modern convergence theorems of iterative schemes for nonexpansive mappings are presented and the main results in the paper generalize and improve the corresponding results given by many authors.